How can I animate an object fading out?
You just need to add another Mix Shader node mixing the output of the first Mix Shader node with another transparency shader. Then you can keyframe the Fac amount on the new Mix Shader node to fade out the object (or more precisely to fade out this material wherever it is applied to the object).

As an extra note, you may want to control the speed of the fade by editing the Fac value transition in the Graph editor.